In order to test some implementations of numerical solvers for advection-diffusion equations with non-constant coefficients, I'm looking for examples of equations+border and initial conditions of this type which have explicit solutions. Could you propose any or references to such?
The method of manufactured solution is very simple. If you have your system $F(u)=0$, then you let your solution be e.g. $u=\sin(x^2)+\cos(\exp(y))$, plug into $F$ to get $F(u)=g(x,y)$. Now $F_0=F(u)-g=0$ is your new equation with solution $u$ and corresponding BC.
I can suggest you the following studies where you can find the exact solution of the numerical examples.
1) Vit Dolejsi, hp-DGFEM for nonlinear convection-diffusion problems, http://dx.doi.org/10.1016/j.matcom.2013.03.001
2) Melanie Bittl, Dmitri Kuzmin, Roland Becker, The CG1-DG2 method for convection–diffusion equations in 2D, http://dx.doi.org/10.1016/j.cam.2014.03.008
3) Tie Zhang, Ying Sheng, Superconvergence and gradient recovery for a finite volume element method for solving convection-diffusion equations, DOI: 10.1002/num.21862
I hope they will work...